J. Phys. Chem. 1987,91, 3 11 1-3 113 7r2(a
lo =
+ b)
In [32a/(b - a) + exp(7r2/4)1
(35)
that was devised in an attempt to describe ring electrodes with arbitrary thickness. It can be seen that eq 35 is exact for a disk and is highly accurate over the whole available range of b/a. The exact “thin-ring” lo in eq 34 is an excellent approximation for b / a C 1.25. Our values of lo (eq 3 1b) are within 2.5%-1.5% of the exact values and improve as the ring becomes thinner. Recently Fleischmann et al.15 studied the steady-state behavior of the current at thin microring electrodes. Their assumption that the surface of the electrode is “uniformly accessible” seems to be equivalent to our assumption that the flux is uniform. However, because of the approximate nature of their mathematical analysis, their results appear to be somewhat different from ours. For example, they estimate that the ratio of the steady-state diffusion-limited currents at a ring with ( b - a ) / ( b a ) = 0.01 and a disk of radius ( b + a)/2 is 0.7375. Equation 31b predicts that this ratio is 0.6587. Our ratio is rather close to that calculated by using the exact (Le., “nonuniform accessibility”) result in eq 34, namely, 0.6689. We also note that the expression they use (their eq 45) to calculate the exact steady-state current is not quite correct. They obtained this equation from the work of Symanski and Bruckenstein12 but failed to take note of the fact that their definition of r’and Ar is not the same as ri and Ar of Symanski and Bruckenstein. We thus believe that Table I of Fleischmann et al.I5 is misleading. The noteworthy feature of our expression (eq 32) for the long-time behavior of the diffusion-limited current is that the coefficient of the PI2 term is uniquely determined by the steady-state current (Le., by lo). This result does not appear to arise from the use of the “uniform flux” boundary condition. It is remarkable that the exact expression obtained by Shoup and Szabo’ for the long-time behavior of the current at a microdisk electrode of radius b can be recast into the form of eq 32 with lo = 4b. In addition, the current to a hemispherical electrode of radius R is given by eq 32 with lo = 27rR. Therefore, it is tempting to conjecture that the exact long-time behavior of the diffusionlimited current at any microelectrode having a closed surface and lying on an essentially infinite support can be described by the equation
+
lim i(t)/nFDCo* = lo(l 1-00
+ lo/(4n3Dt)’/’)
(36)
(15) Reischmann, M.; Bandyopadhyay,S.;Pons, S. J . Phys. Chem. 1985,
3111
This conjecture can be tested by numerical simulations. For microring electrodes of arbitrary thickness, we suggest that one shoula‘use eq 36 with lo given by eq 35 to analyze experimental data. For arbitrary electrode potentials, we have found that the current at a microring electrode behaves as lim r--
i ( t ) ( k f+ kb) = 1(1 nFD(k,-Co* - k&R*)
+ 1 / ( 4 ~ ~ D t ) ’ (37) /~)
with
-1 = -1+
D
l0 A ( k f + kb) where A is the area of the electrode. We note that these equations describe the exact long-time behavior5 of hemispherical electrodes of radius R when lo = 27rR. It is possible that the functional form of eq 37 is exact; however, eq 38 is an approximation although probably a very good one. The utility of these results is that they express the long-time behavior of the general i-E characteristic in terms of the steady-state diffusion-limited current (Le., lo).
Acknowledgment. I thank Professors A. G. Ewing and R. M. Wightman for useful comments on the manuscript.
Appendix At the request of a referee, we briefly indicate here how eq 30 was obtained from eq 29. We first note that eq 29 can be rewritten as F(0) = -[4(b3 1 37r
+ a3)- ( a + b)((a+ b)’E(k)
-
(b - a)2(2K(k) - E(k))11 (AI)
+
Now since k2 = 4nb/(a b)2,k goes to unity as ( b - a) approaches zero. In this limit16 K ( k ) = In (4/k’) + ... (‘42) E(k) = 1
+ Y2(k’)’(ln
(4/k’) -
Y2) + ...
(A3)
where
Using eq A2, A3, and A4 in eq A l , we recover eq 30 of the text. (16) Gradsteyn, I. S.; Ryzhik, I. M. Table of Integrals, Series and ProdNew York, 1980; p 905.
ucts; Academic:
89, 5537.
COMMENTS Dielectric Spectrum of a 0.5 M Aqueous NaCi Solution Sir: The principal dielectric relaxation time T~ of aqueous solutions of small inorganic ions is frequently smaller than that of pure water, T,, a t the same temperature.’” Different molecular (1) Giese, K.; Kaatze, U.; Pottel, R. J . Phys. Chem. 1970, 74, 3718. (2) Barthel, J.; Behret, H.; Schmithals, F. Eer. Bunsenges. Phys. Chem. 1971, 75, 305.
(3) Pottel, R. In Water, a Comprehensiue Treatise; Franks, F. Ed.;Plenum: New York, 1973; Vol. 3, p 401. (4) Hasted, J. B. Aqueous Dielectrics; Chapman and Hall: London, 1973; p 136.
0022-3654/87/2091-3111$01.50/0
mechanisms have been discussed to explain this finding, among them the effects of kinetic depolarization,6dielectric negative hydration:-” and hydration water exchange.12 But a
(5) Kaatze, U. 2.Phys. Chem. (Miinchen) 1983, 135, 51. (6) Hubbard, J. B. J . Chem. Phys. 1978, 68, 1649. (7) Hubbard, J. B.; Onsager, L. J . Chem. Phys. 1977, 67, 4850. (8) Ibuki, K.; Nakahara, M. J . Chem. Phys. 1986, 84, 2776. (9) Samoilov, 0. Ya Discuss. Faraday SOC.1957, 24, 141. (IO) Engel, G.; Hertz, H. G. Ber. Bunsenges. Phys. Chem. 1968, 72, 808. (1 1) Pottel, R.; Giese, K.; Kaatze, U. In Structure of Waferand Aqueous Solutions; Luck, W. A. P., Ed.; Verlag Chemie, Physik Verlag: Weinheim, 1974; p 391.
0 1987 American Chemical Society
3112 The Journal of Physical Chemistry, Vol. 91, No, 11, 1987
Comments 101
0 99
60 0 97 40
0 95 rf . 093
20
0 91
0 I
I
0 89
I
0 81
02
0
04
0 6 mollkg 1 0 m
Figure 2. Relaxation time ratio r , / r . as a function of the molal concentration m of solute for aqueous NaCl solutions (closed circle, this paper; half-closed circles, previous results from this laboratory at 25 O C ; ' open circles, data from the literature for 25 OCzwzz and 20 "CZ3).
80
I
I
'
I
I
I
I
I
1
'2
3
4
5
I
I
I
I
I
1
70 1
I
I
1
3
10
30 GHz
I 100
60
V
-u
Figure 1. Real part, c'(v), and negative imaginary part excluding conductivity contributions, e"@) - u / ( 2 m 0 v ) ,of the complex permittivity plotted as a function of frequency v for water (crosses, ref 18) and the 0.5 M aqueous solution of NaCl (closed circles) at 25 "C. For the solution the lines represent the Cole-Cole relaxation function (eq 1) with the parameter values found by the fitting procedure.
satisfactory comprehensive model description has not been found so far. Recently, Winsor and Cole, when evaluating time-domain spectroscopic (TDS) data for 0.05,0.1, and 0.5 M aqueous NaCl solutions, derived dramatically increased relaxation times ( T , / T ~ i= 1.5 at c = 0.5 mol/Li3). Until then such a pronounced positive shift in the T,-value of aqueous electrolyte solutions had been only found if strongly hydrophobic ions were in~olved.'~ In this situation it seemed to be necessary to try to accurately remeasure the dielectric spectrum of a 0.5 M aqueous NaCl solution over a frequency range as broad as possible and to compare the results with previous frequency-domain data and the recent TDS data as well. Spot frequency measurements according to the travelling wave method first proposed by BuchananlS have been performed for this purpose. This method involves a double beam interferometer, the wave transmitted through a specimen cell of variable length being balanced against a reference wave of fixed phase and calibrated variable amplitude. Five microwave bridge circuits were utilized to cover the frequency range from 1 to 40 G H z . ' ~ ~ " Altogether seven sample cells and ten different attenuators were used to minimize a n y effects of possible systematic errors by undesired spurious modes in the cells and by imperfections of the microwave apparatus. The specific electric conductivity CT of the solution has been measured in the usual manner at 1, 10, and 100 kHz. The measured permittivity data reduced by the conductivity contribution iu/(2mov) are presented in Figure 1 where for reasons of comparison the dielectric spectrum of pure water at 25 OC is (12) Giese, K. Ber. Bunsenges. Phys. Chem. 1972, 76, 495. (13) Winsor, P.; Cole, R. H. J . Phys. Chem. 1985, 89, 3775. (14) Wen, W.-Y.; Kaatze, U. J. Phys. Chem. 1977, 81, 177. (15) Buchanan, T. J. Proc. IEE 1952, 99, 61. (16) Kaatze, U. Adv. Mol. Relaxation Processes 1975, 7 , 45. (17) Kaatze, U. Mikrowellen Mag. 1980, 46.
50 40
30 20 10
0
I
lo'%-'
7
2rrVh"(VI - d E 0
Figure 3. Plot of d ( v ) vs. 27rve"(u) - u/co for a 0.5 M aqueous NaCl solution at 25 "C (closed circles, this paper; open circles, TDS result^'^). The TDS data have been taken with a ruler from the lower part of Figure 1 in the paper of Winsor and C01e.l~
also displayed (e0 = 8.854 X lo-'* A s V-' m-I). If a Cole-Cole relaxation spectral function19 given by the relation e(v)
CT + i-27Tcov = e'(v) - i
is fitted to the data of the solution, the following values are found for the parameters: e(0) = 72.4 f 0.7; €(a)= 4.4 f 1; T~ = 7.82 f 0.15 ps; h = 0.04 f 0.02. These values agree with those obtained if eq 1 is fitted to the previous (reduced) set of permittivity data for the 0.5 M aqueous N a C l solution:' e(0) = 72.8 1; e(=) = 4.5 f 1; 7 , = 7.85 f 0.2 ps; h = 0.03 f 0.02. Within the limits of experimental error the T , value also fits to other relaxation time values previously published on the basis of frequency-domain data for aqueous NaCl solutions with concentrations up to 1 M (Figure 2). (18) Kaatze, U.; Uhlendorf, V. 2.Phys. Chem. (Wiesboden) 1981, 129, 151. (19) Cole, K. S.; Cole, R. H. J. Chem. Phys. 1941, 9, 341. (20) Haggis, G. H.; Hasted, J. B.; Buchanan, T. J. J . Chem. Phys. 1952, 20, 1452. (21) Hasted, J. B.; Roderick, G. W. J. Chem. Phys. 1958, 29, 17. (22) Schmithals, F. Ph.D. Thesis, University of Saarbriicken, 1969. (23) Hasted, J. B.; El Sabeh, S. H. M. Trans. Faraday Soc. 1953, 49, 1003.
J . Phys. Chem. 1987, 91, 3113-3114
To throw some light on the question why different relaxation time values result from the recent TDS measurements, ef is displayed as a function of 2 7 ” - u/tOin Figure 3. Winsor and Cole used this type of plot to determine the relaxation times of the solutions. In the case of a Debye relaxationz4the plot of e’ vs. (2rvt” - a/eo) is a straight line and the negative slope of it equals the relaxation time.2s The e’ vs. (2nvt” - u/eo) relation shows a small curvature indicating that there is a more complex than the Debye behavior. As a consequence of the curvature the initial negative slope of the plot somewhat exceeds the principal relaxation time 7,. Additionally, however, it is found that the e’ values derived from the TDS study show a greater decrease with increasing ve” than the present data do in the lower frequency band (v 5 7 GHz, Figure 3). This discrepancy may probably be due to difficulties in the measurements of highly conducting samples by fast response time-domain techniques. The empirical Cole-Cole relaxation spectral function (eq 1) has been used here to analytically represent the measured dielectric spectrum of NaCl solutions for the following reasons. Above all, this function allows the commonly used principal relaxation time 7,to be found by interpolation. The principal relaxation time is defined by the frequency us = (2n~,)-’at which the dielectric losses (Figure 1) adopt the relative maximum: d(df(v,) - ~ / ( 2 ~ t o ~ , ) )=/ d0 ~ dZ(d’(v,) - a / ( 2 ~ e o ~ , ) ) / d v (2~7,)-’, Figure 1) is taken into account. Unfortunately, an easy interpretation of the initial slope in the e’ vs. (2nvt” - a/€,,) relation is not possible in the case of an underlying relaxation time distribution (including a superposition of Debye terms if their relaxation times have similar values). This slope equals the negative value of a relaxation time only if a discrete process or a sum of well-separated Debye-type relaxations is considered. For this reason the present evaluation of measurements has been restricted to the analysis of the original dispersion and dielectric loss function. Registry No. NaCI, 7647-14-5. (24) Debye, P. Polare Molekeln; Hirzel: Leipzig, 1929. (25) Cole, R. H. J . Chem. Phys. 1955, 23, 493.
Drittes Physikafisches Institut Universitat Gottingen 0-3400 Gottingen, West Germany
Udo Kaatze
Received: September 5, 1986; In Final Form: January 21, 1987 0022-3654/87/2091-3113$01.50/0
3113
Comment on “Electron-Transfer Reactlons of Tryptophan and Tyrosine Derivatives” Sir: A recent paper’ confirms that, in neutral solutions, radicals obtained by removal of electrons from tryptophan derivatives are able to transfer their electron deficiencies to tyrosine derivatives? OH
R
H
R
One-electron reduction potentials of the oxidized radicals of tryptophan, tyrosine, and some derivatives have been measured a t three widely separated pH values.’ The potentials for the derivatives were little different from those for the parents. At pH 3, 7, and 13 the values for tryptophan itself were given as 0.94, 0.64, and 0.56 V (vs. NHE), respectively. The value at pH 7 was measured with p-methoxyphenol as reference standard. It compares with potentials at this p H close to 0.85 V for the radicals of tyrosine and derivatives,’ so that the transfer would have been expected to be thermodynamically forbidden. In view of the inconsistency, an ingenious mechanism involving a complex between the ring nitrogen of the tryptophan radical and the hydroxyl group of tyrosine was proposed to account for the supposedly forbidden reaction.’ We have three principal criticisms of the paper: (1) If the side chains’in the oxidized tryptophan radical were to have the same pK, as the side chains in the parent, as implied for tyrosine by the use of formula 15 of ref 1 , the one-electron reduction potential, Ei, of the oxidized tryptophan radical would vary with pH, i, according to the following formula, where 4.3 is the pK, of the protonated form of the oxidized tryptophan radical3 and Eo is the one-electron reduction potential at pH 0:
This formula would predict values of 0.78 and 0.91 V for pH 7 from the experimental values given for pH 3 and 13, respectively.’ The value of 0.78 would be insensitive to any difference between the pKa of the side chains of the parent and the radical. The value of 0.91 V could become as low as 0.81 V if the pKa of the protonated amino group in the radical were to be as high as 1 1 . 1 (cf. 9.39 for the parent) or as high as 1.01 V if the pK, were to be as low as 7.7. All these values are consistent with the published values of 0.98 f 0.22 and 0.87 f 0.1 V,4 whereas the reported’ value of 0.64 V is inconsistent with all other data. (2) There is a fallacy in the proposal of the mechanism to account for the supposedly forbidden reaction. If complex formation between the ring nitrogen of tryptophan and the hydroxyl group of tyrosine could cause the transfer to proceed against thermodynamic equilibrium, then complex formation with the hydroxyl group of p-methoxyphenol could also cause transfer against thermodynamic equilibrium and the one-electron reduction potential of the oxidized tryptophan radical would not be valid. There would then be no reason to doubt that the potential was above that for the tyrosine radical, and there would be no reason to propose the mechanism! No conceivable alternative reaction mechanism could enable this reaction to proceed either, if it were genuinely thermodynamically forbidden. ( 1 ) Jovanovic, S.V.; Harriman, A,; Simic, M. G.J . Phys. Chem. 1986, 90, 1935-1939. (2) Butler, J.; Land, E. J.; Prutz, W. A.; Swallow, A. J. Eiochim. Biophys. . . Acta 1982, 705, 150-162. (3) Posener, M. L.; Adams, G.E.; Wardman, P.; Cundall, R. B. J . Chem. Soc., Faraday Trans. 1 1976, 72, 2231-2239.
0 1987 American Chemical Society
